LMFDB - Embedded newform 8624.2.a.cd.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8624,2,Mod(1,8624)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8624.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8624, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 8624 = 2^{4} \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8624.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,2,0,0,0,0,0,0,0,-2,0,2,0,4,0,4,0,-8,0,0,0,-8,0,-6,0,2,0, 2,0,-8,0,-2,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(37)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$68.8629867032$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 616)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	8624.1

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.414214 q^{3} -1.41421 q^{5} -2.82843 q^{9} -1.00000 q^{11} -1.82843 q^{13} +0.585786 q^{15} +2.00000 q^{17} -2.58579 q^{19} -5.41421 q^{23} -3.00000 q^{25} +2.41421 q^{27} +1.00000 q^{29} -9.65685 q^{31} +0.414214 q^{33} -9.07107 q^{37} +0.757359 q^{39} -6.24264 q^{41} +8.00000 q^{43} +4.00000 q^{45} +4.82843 q^{47} -0.828427 q^{51} +2.58579 q^{53} +1.41421 q^{55} +1.07107 q^{57} +4.41421 q^{59} +8.17157 q^{61} +2.58579 q^{65} -8.07107 q^{67} +2.24264 q^{69} +5.75736 q^{71} +3.41421 q^{73} +1.24264 q^{75} -8.07107 q^{79} +7.48528 q^{81} -10.4853 q^{83} -2.82843 q^{85} -0.414214 q^{87} +9.17157 q^{89} +4.00000 q^{93} +3.65685 q^{95} +15.8284 q^{97} +2.82843 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 2 q^{11} + 2 q^{13} + 4 q^{15} + 4 q^{17} - 8 q^{19} - 8 q^{23} - 6 q^{25} + 2 q^{27} + 2 q^{29} - 8 q^{31} - 2 q^{33} - 4 q^{37} + 10 q^{39} - 4 q^{41} + 16 q^{43} + 8 q^{45} + 4 q^{47}+ \cdots + 26 q^{97}+O(q^{100}) $ 2 * q + 2 * q^3 - 2 * q^11 + 2 * q^13 + 4 * q^15 + 4 * q^17 - 8 * q^19 - 8 * q^23 - 6 * q^25 + 2 * q^27 + 2 * q^29 - 8 * q^31 - 2 * q^33 - 4 * q^37 + 10 * q^39 - 4 * q^41 + 16 * q^43 + 8 * q^45 + 4 * q^47 + 4 * q^51 + 8 * q^53 - 12 * q^57 + 6 * q^59 + 22 * q^61 + 8 * q^65 - 2 * q^67 - 4 * q^69 + 20 * q^71 + 4 * q^73 - 6 * q^75 - 2 * q^79 - 2 * q^81 - 4 * q^83 + 2 * q^87 + 24 * q^89 + 8 * q^93 - 4 * q^95 + 26 * q^97

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−0.414214	−0.239146	−0.119573	−	0.992825i	$-0.538153\pi$
$3$	−0.414214	−0.239146	−0.119573	+	0.992825i	$0.538153\pi$
$4$	0	0
$5$	−1.41421	−0.632456	−0.316228	−	0.948683i	$-0.602416\pi$
$5$	−1.41421	−0.632456	−0.316228	+	0.948683i	$0.602416\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−2.82843	−0.942809
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−1.82843	−0.507114	−0.253557	−	0.967320i	$-0.581601\pi$
$13$	−1.82843	−0.507114	−0.253557	+	0.967320i	$0.581601\pi$
$14$	0	0
$15$	0.585786	0.151249
$16$	0	0
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	−2.58579	−0.593220	−0.296610	−	0.954999i	$-0.595856\pi$
$19$	−2.58579	−0.593220	−0.296610	+	0.954999i	$0.595856\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−5.41421	−1.12894	−0.564471	−	0.825453i	$-0.690920\pi$
$23$	−5.41421	−1.12894	−0.564471	+	0.825453i	$0.690920\pi$
$24$	0	0
$25$	−3.00000	−0.600000
$26$	0	0
$27$	2.41421	0.464616
$28$	0	0
$29$	1.00000	0.185695	0.0928477	−	0.995680i	$-0.470403\pi$
$29$	1.00000	0.185695	0.0928477	+	0.995680i	$0.470403\pi$
$30$	0	0
$31$	−9.65685	−1.73442	−0.867211	−	0.497941i	$-0.834090\pi$
$31$	−9.65685	−1.73442	−0.867211	+	0.497941i	$0.834090\pi$
$32$	0	0
$33$	0.414214	0.0721053
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−9.07107	−1.49127	−0.745637	−	0.666352i	$-0.767855\pi$
$37$	−9.07107	−1.49127	−0.745637	+	0.666352i	$0.767855\pi$
$38$	0	0
$39$	0.757359	0.121275
$40$	0	0
$41$	−6.24264	−0.974937	−0.487468	−	0.873141i	$-0.662080\pi$
$41$	−6.24264	−0.974937	−0.487468	+	0.873141i	$0.662080\pi$
$42$	0	0
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	0	0
$45$	4.00000	0.596285
$46$	0	0
$47$	4.82843	0.704298	0.352149	−	0.935944i	$-0.385451\pi$
$47$	4.82843	0.704298	0.352149	+	0.935944i	$0.385451\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−0.828427	−0.116003
$52$	0	0
$53$	2.58579	0.355185	0.177593	−	0.984104i	$-0.443169\pi$
$53$	2.58579	0.355185	0.177593	+	0.984104i	$0.443169\pi$
$54$	0	0
$55$	1.41421	0.190693
$56$	0	0
$57$	1.07107	0.141866
$58$	0	0
$59$	4.41421	0.574682	0.287341	−	0.957828i	$-0.407229\pi$
$59$	4.41421	0.574682	0.287341	+	0.957828i	$0.407229\pi$
$60$	0	0
$61$	8.17157	1.04626	0.523131	−	0.852252i	$-0.324764\pi$
$61$	8.17157	1.04626	0.523131	+	0.852252i	$0.324764\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	2.58579	0.320727
$66$	0	0
$67$	−8.07107	−0.986038	−0.493019	−	0.870019i	$-0.664107\pi$
$67$	−8.07107	−0.986038	−0.493019	+	0.870019i	$0.664107\pi$
$68$	0	0
$69$	2.24264	0.269982
$70$	0	0
$71$	5.75736	0.683273	0.341636	−	0.939832i	$-0.389019\pi$
$71$	5.75736	0.683273	0.341636	+	0.939832i	$0.389019\pi$
$72$	0	0
$73$	3.41421	0.399603	0.199802	−	0.979836i	$-0.435970\pi$
$73$	3.41421	0.399603	0.199802	+	0.979836i	$0.435970\pi$
$74$	0	0
$75$	1.24264	0.143488
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−8.07107	−0.908066	−0.454033	−	0.890985i	$-0.650015\pi$
$79$	−8.07107	−0.908066	−0.454033	+	0.890985i	$0.650015\pi$
$80$	0	0
$81$	7.48528	0.831698
$82$	0	0
$83$	−10.4853	−1.15091	−0.575455	−	0.817834i	$-0.695175\pi$
$83$	−10.4853	−1.15091	−0.575455	+	0.817834i	$0.695175\pi$
$84$	0	0
$85$	−2.82843	−0.306786
$86$	0	0
$87$	−0.414214	−0.0444084
$88$	0	0
$89$	9.17157	0.972185	0.486092	−	0.873907i	$-0.338422\pi$
$89$	9.17157	0.972185	0.486092	+	0.873907i	$0.338422\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	4.00000	0.414781
$94$	0	0
$95$	3.65685	0.375185
$96$	0	0
$97$	15.8284	1.60713	0.803567	−	0.595215i	$-0.202933\pi$
$97$	15.8284	1.60713	0.803567	+	0.595215i	$0.202933\pi$
$98$	0	0
$99$	2.82843	0.284268
$100$	0	0
$101$	7.48528	0.744813	0.372407	−	0.928070i	$-0.378533\pi$
$101$	7.48528	0.744813	0.372407	+	0.928070i	$0.378533\pi$
$102$	0	0
$103$	−15.8995	−1.56662	−0.783312	−	0.621629i	$-0.786471\pi$
$103$	−15.8995	−1.56662	−0.783312	+	0.621629i	$0.786471\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−13.5563	−1.31054	−0.655271	−	0.755394i	$-0.727446\pi$
$107$	−13.5563	−1.31054	−0.655271	+	0.755394i	$0.727446\pi$
$108$	0	0
$109$	−8.48528	−0.812743	−0.406371	−	0.913708i	$-0.633206\pi$
$109$	−8.48528	−0.812743	−0.406371	+	0.913708i	$0.633206\pi$
$110$	0	0
$111$	3.75736	0.356633
$112$	0	0
$113$	−7.48528	−0.704156	−0.352078	−	0.935971i	$-0.614525\pi$
$113$	−7.48528	−0.704156	−0.352078	+	0.935971i	$0.614525\pi$
$114$	0	0
$115$	7.65685	0.714005
$116$	0	0
$117$	5.17157	0.478112
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	2.58579	0.233153
$124$	0	0
$125$	11.3137	1.01193
$126$	0	0
$127$	−6.41421	−0.569169	−0.284585	−	0.958651i	$-0.591856\pi$
$127$	−6.41421	−0.569169	−0.284585	+	0.958651i	$0.591856\pi$
$128$	0	0
$129$	−3.31371	−0.291756
$130$	0	0
$131$	−4.92893	−0.430643	−0.215321	−	0.976543i	$-0.569080\pi$
$131$	−4.92893	−0.430643	−0.215321	+	0.976543i	$0.569080\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−3.41421	−0.293849
$136$	0	0
$137$	17.0000	1.45241	0.726204	−	0.687479i	$-0.241283\pi$
$137$	17.0000	1.45241	0.726204	+	0.687479i	$0.241283\pi$
$138$	0	0
$139$	−3.31371	−0.281065	−0.140533	−	0.990076i	$-0.544881\pi$
$139$	−3.31371	−0.281065	−0.140533	+	0.990076i	$0.544881\pi$
$140$	0	0
$141$	−2.00000	−0.168430
$142$	0	0
$143$	1.82843	0.152901
$144$	0	0
$145$	−1.41421	−0.117444
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−16.0000	−1.31077	−0.655386	−	0.755295i	$-0.727494\pi$
$149$	−16.0000	−1.31077	−0.655386	+	0.755295i	$0.727494\pi$
$150$	0	0
$151$	−5.58579	−0.454565	−0.227282	−	0.973829i	$-0.572984\pi$
$151$	−5.58579	−0.454565	−0.227282	+	0.973829i	$0.572984\pi$
$152$	0	0
$153$	−5.65685	−0.457330
$154$	0	0
$155$	13.6569	1.09694
$156$	0	0
$157$	7.31371	0.583697	0.291849	−	0.956464i	$-0.405730\pi$
$157$	7.31371	0.583697	0.291849	+	0.956464i	$0.405730\pi$
$158$	0	0
$159$	−1.07107	−0.0849412
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−5.10051	−0.399502	−0.199751	−	0.979847i	$-0.564013\pi$
$163$	−5.10051	−0.399502	−0.199751	+	0.979847i	$0.564013\pi$
$164$	0	0
$165$	−0.585786	−0.0456034
$166$	0	0
$167$	−0.272078	−0.0210540	−0.0105270	−	0.999945i	$-0.503351\pi$
$167$	−0.272078	−0.0210540	−0.0105270	+	0.999945i	$0.503351\pi$
$168$	0	0
$169$	−9.65685	−0.742835
$170$	0	0
$171$	7.31371	0.559293
$172$	0	0
$173$	−2.17157	−0.165102	−0.0825508	−	0.996587i	$-0.526307\pi$
$173$	−2.17157	−0.165102	−0.0825508	+	0.996587i	$0.526307\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−1.82843	−0.137433
$178$	0	0
$179$	−18.8995	−1.41261	−0.706307	−	0.707905i	$-0.749640\pi$
$179$	−18.8995	−1.41261	−0.706307	+	0.707905i	$0.749640\pi$
$180$	0	0
$181$	13.3137	0.989600	0.494800	−	0.869007i	$-0.335241\pi$
$181$	13.3137	0.989600	0.494800	+	0.869007i	$0.335241\pi$
$182$	0	0
$183$	−3.38478	−0.250210
$184$	0	0
$185$	12.8284	0.943165
$186$	0	0
$187$	−2.00000	−0.146254
$188$	0	0
$189$	0	0
$190$	0	0
$191$	6.48528	0.469258	0.234629	−	0.972085i	$-0.424612\pi$
$191$	6.48528	0.469258	0.234629	+	0.972085i	$0.424612\pi$
$192$	0	0
$193$	23.2132	1.67092	0.835461	−	0.549549i	$-0.185200\pi$
$193$	23.2132	1.67092	0.835461	+	0.549549i	$0.185200\pi$
$194$	0	0
$195$	−1.07107	−0.0767008
$196$	0	0
$197$	−11.1421	−0.793844	−0.396922	−	0.917852i	$-0.629922\pi$
$197$	−11.1421	−0.793844	−0.396922	+	0.917852i	$0.629922\pi$
$198$	0	0
$199$	24.0416	1.70427	0.852133	−	0.523325i	$-0.175309\pi$
$199$	24.0416	1.70427	0.852133	+	0.523325i	$0.175309\pi$
$200$	0	0
$201$	3.34315	0.235807
$202$	0	0
$203$	0	0
$204$	0	0
$205$	8.82843	0.616604
$206$	0	0
$207$	15.3137	1.06438
$208$	0	0
$209$	2.58579	0.178863
$210$	0	0
$211$	4.92893	0.339322	0.169661	−	0.985503i	$-0.445733\pi$
$211$	4.92893	0.339322	0.169661	+	0.985503i	$0.445733\pi$
$212$	0	0
$213$	−2.38478	−0.163402
$214$	0	0
$215$	−11.3137	−0.771589
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−1.41421	−0.0955637
$220$	0	0
$221$	−3.65685	−0.245987
$222$	0	0
$223$	19.5563	1.30959	0.654795	−	0.755807i	$-0.272755\pi$
$223$	19.5563	1.30959	0.654795	+	0.755807i	$0.272755\pi$
$224$	0	0
$225$	8.48528	0.565685
$226$	0	0
$227$	−28.1421	−1.86786	−0.933930	−	0.357457i	$-0.883644\pi$
$227$	−28.1421	−1.86786	−0.933930	+	0.357457i	$0.883644\pi$
$228$	0	0
$229$	4.00000	0.264327	0.132164	−	0.991228i	$-0.457808\pi$
$229$	4.00000	0.264327	0.132164	+	0.991228i	$0.457808\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	20.3848	1.33545	0.667726	−	0.744408i	$-0.267268\pi$
$233$	20.3848	1.33545	0.667726	+	0.744408i	$0.267268\pi$
$234$	0	0
$235$	−6.82843	−0.445437
$236$	0	0
$237$	3.34315	0.217161
$238$	0	0
$239$	11.7279	0.758616	0.379308	−	0.925270i	$-0.376162\pi$
$239$	11.7279	0.758616	0.379308	+	0.925270i	$0.376162\pi$
$240$	0	0
$241$	22.7279	1.46403	0.732017	−	0.681286i	$-0.238579\pi$
$241$	22.7279	1.46403	0.732017	+	0.681286i	$0.238579\pi$
$242$	0	0
$243$	−10.3431	−0.663513
$244$	0	0
$245$	0	0
$246$	0	0
$247$	4.72792	0.300830
$248$	0	0
$249$	4.34315	0.275236
$250$	0	0
$251$	0.485281	0.0306307	0.0153153	−	0.999883i	$-0.495125\pi$
$251$	0.485281	0.0306307	0.0153153	+	0.999883i	$0.495125\pi$
$252$	0	0
$253$	5.41421	0.340389
$254$	0	0
$255$	1.17157	0.0733667
$256$	0	0
$257$	19.4853	1.21546	0.607729	−	0.794144i	$-0.292081\pi$
$257$	19.4853	1.21546	0.607729	+	0.794144i	$0.292081\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−2.82843	−0.175075
$262$	0	0
$263$	23.0416	1.42081	0.710404	−	0.703794i	$-0.248512\pi$
$263$	23.0416	1.42081	0.710404	+	0.703794i	$0.248512\pi$
$264$	0	0
$265$	−3.65685	−0.224639
$266$	0	0
$267$	−3.79899	−0.232494
$268$	0	0
$269$	13.6569	0.832673	0.416337	−	0.909211i	$-0.363314\pi$
$269$	13.6569	0.832673	0.416337	+	0.909211i	$0.363314\pi$
$270$	0	0
$271$	−0.0710678	−0.00431706	−0.00215853	−	0.999998i	$-0.500687\pi$
$271$	−0.0710678	−0.00431706	−0.00215853	+	0.999998i	$0.500687\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	3.00000	0.180907
$276$	0	0
$277$	−24.7990	−1.49003	−0.745013	−	0.667049i	$-0.767557\pi$
$277$	−24.7990	−1.49003	−0.745013	+	0.667049i	$0.767557\pi$
$278$	0	0
$279$	27.3137	1.63523
$280$	0	0
$281$	−11.8995	−0.709864	−0.354932	−	0.934892i	$-0.615496\pi$
$281$	−11.8995	−0.709864	−0.354932	+	0.934892i	$0.615496\pi$
$282$	0	0
$283$	27.5563	1.63805	0.819027	−	0.573754i	$-0.194514\pi$
$283$	27.5563	1.63805	0.819027	+	0.573754i	$0.194514\pi$
$284$	0	0
$285$	−1.51472	−0.0897242
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	−6.55635	−0.384340
$292$	0	0
$293$	13.1716	0.769492	0.384746	−	0.923023i	$-0.374289\pi$
$293$	13.1716	0.769492	0.384746	+	0.923023i	$0.374289\pi$
$294$	0	0
$295$	−6.24264	−0.363461
$296$	0	0
$297$	−2.41421	−0.140087
$298$	0	0
$299$	9.89949	0.572503
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−3.10051	−0.178119
$304$	0	0
$305$	−11.5563	−0.661715
$306$	0	0
$307$	8.58579	0.490017	0.245008	−	0.969521i	$-0.421209\pi$
$307$	8.58579	0.490017	0.245008	+	0.969521i	$0.421209\pi$
$308$	0	0
$309$	6.58579	0.374652
$310$	0	0
$311$	−27.2132	−1.54312	−0.771560	−	0.636157i	$-0.780523\pi$
$311$	−27.2132	−1.54312	−0.771560	+	0.636157i	$0.780523\pi$
$312$	0	0
$313$	9.34315	0.528106	0.264053	−	0.964508i	$-0.414941\pi$
$313$	9.34315	0.528106	0.264053	+	0.964508i	$0.414941\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	20.9706	1.17782	0.588912	−	0.808197i	$-0.299556\pi$
$317$	20.9706	1.17782	0.588912	+	0.808197i	$0.299556\pi$
$318$	0	0
$319$	−1.00000	−0.0559893
$320$	0	0
$321$	5.61522	0.313411
$322$	0	0
$323$	−5.17157	−0.287754
$324$	0	0
$325$	5.48528	0.304269
$326$	0	0
$327$	3.51472	0.194364
$328$	0	0
$329$	0	0
$330$	0	0
$331$	5.38478	0.295974	0.147987	−	0.988989i	$-0.452721\pi$
$331$	5.38478	0.295974	0.147987	+	0.988989i	$0.452721\pi$
$332$	0	0
$333$	25.6569	1.40599
$334$	0	0
$335$	11.4142	0.623625
$336$	0	0
$337$	−2.24264	−0.122164	−0.0610822	−	0.998133i	$-0.519455\pi$
$337$	−2.24264	−0.122164	−0.0610822	+	0.998133i	$0.519455\pi$
$338$	0	0
$339$	3.10051	0.168396
$340$	0	0
$341$	9.65685	0.522948
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−3.17157	−0.170752
$346$	0	0
$347$	8.58579	0.460909	0.230455	−	0.973083i	$-0.425979\pi$
$347$	8.58579	0.460909	0.230455	+	0.973083i	$0.425979\pi$
$348$	0	0
$349$	−24.2843	−1.29991	−0.649954	−	0.759974i	$-0.725212\pi$
$349$	−24.2843	−1.29991	−0.649954	+	0.759974i	$0.725212\pi$
$350$	0	0
$351$	−4.41421	−0.235613
$352$	0	0
$353$	2.68629	0.142977	0.0714884	−	0.997441i	$-0.477225\pi$
$353$	2.68629	0.142977	0.0714884	+	0.997441i	$0.477225\pi$
$354$	0	0
$355$	−8.14214	−0.432140
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−13.9289	−0.735141	−0.367570	−	0.929996i	$-0.619810\pi$
$359$	−13.9289	−0.735141	−0.367570	+	0.929996i	$0.619810\pi$
$360$	0	0
$361$	−12.3137	−0.648090
$362$	0	0
$363$	−0.414214	−0.0217406
$364$	0	0
$365$	−4.82843	−0.252731
$366$	0	0
$367$	8.24264	0.430262	0.215131	−	0.976585i	$-0.430982\pi$
$367$	8.24264	0.430262	0.215131	+	0.976585i	$0.430982\pi$
$368$	0	0
$369$	17.6569	0.919179
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−13.6274	−0.705601	−0.352800	−	0.935699i	$-0.614771\pi$
$373$	−13.6274	−0.705601	−0.352800	+	0.935699i	$0.614771\pi$
$374$	0	0
$375$	−4.68629	−0.241999
$376$	0	0
$377$	−1.82843	−0.0941688
$378$	0	0
$379$	7.38478	0.379330	0.189665	−	0.981849i	$-0.439260\pi$
$379$	7.38478	0.379330	0.189665	+	0.981849i	$0.439260\pi$
$380$	0	0
$381$	2.65685	0.136115
$382$	0	0
$383$	−8.10051	−0.413916	−0.206958	−	0.978350i	$-0.566356\pi$
$383$	−8.10051	−0.413916	−0.206958	+	0.978350i	$0.566356\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−22.6274	−1.15022
$388$	0	0
$389$	33.2132	1.68398	0.841988	−	0.539496i	$-0.181385\pi$
$389$	33.2132	1.68398	0.841988	+	0.539496i	$0.181385\pi$
$390$	0	0
$391$	−10.8284	−0.547617
$392$	0	0
$393$	2.04163	0.102987
$394$	0	0
$395$	11.4142	0.574311
$396$	0	0
$397$	−35.6569	−1.78957	−0.894783	−	0.446501i	$-0.852670\pi$
$397$	−35.6569	−1.78957	−0.894783	+	0.446501i	$0.852670\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−25.0000	−1.24844	−0.624220	−	0.781248i	$-0.714583\pi$
$401$	−25.0000	−1.24844	−0.624220	+	0.781248i	$0.714583\pi$
$402$	0	0
$403$	17.6569	0.879551
$404$	0	0
$405$	−10.5858	−0.526012
$406$	0	0
$407$	9.07107	0.449636
$408$	0	0
$409$	33.8995	1.67622	0.838111	−	0.545500i	$-0.183660\pi$
$409$	33.8995	1.67622	0.838111	+	0.545500i	$0.183660\pi$
$410$	0	0
$411$	−7.04163	−0.347338
$412$	0	0
$413$	0	0
$414$	0	0
$415$	14.8284	0.727899
$416$	0	0
$417$	1.37258	0.0672157
$418$	0	0
$419$	−25.4558	−1.24360	−0.621800	−	0.783176i	$-0.713598\pi$
$419$	−25.4558	−1.24360	−0.621800	+	0.783176i	$0.713598\pi$
$420$	0	0
$421$	9.65685	0.470646	0.235323	−	0.971917i	$-0.424385\pi$
$421$	9.65685	0.470646	0.235323	+	0.971917i	$0.424385\pi$
$422$	0	0
$423$	−13.6569	−0.664019
$424$	0	0
$425$	−6.00000	−0.291043
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−0.757359	−0.0365657
$430$	0	0
$431$	−7.72792	−0.372241	−0.186120	−	0.982527i	$-0.559591\pi$
$431$	−7.72792	−0.372241	−0.186120	+	0.982527i	$0.559591\pi$
$432$	0	0
$433$	13.4558	0.646647	0.323323	−	0.946289i	$-0.395200\pi$
$433$	13.4558	0.646647	0.323323	+	0.946289i	$0.395200\pi$
$434$	0	0
$435$	0.585786	0.0280863
$436$	0	0
$437$	14.0000	0.669711
$438$	0	0
$439$	1.92893	0.0920629	0.0460315	−	0.998940i	$-0.485343\pi$
$439$	1.92893	0.0920629	0.0460315	+	0.998940i	$0.485343\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−5.31371	−0.252462	−0.126231	−	0.992001i	$-0.540288\pi$
$443$	−5.31371	−0.252462	−0.126231	+	0.992001i	$0.540288\pi$
$444$	0	0
$445$	−12.9706	−0.614864
$446$	0	0
$447$	6.62742	0.313466
$448$	0	0
$449$	8.00000	0.377543	0.188772	−	0.982021i	$-0.439549\pi$
$449$	8.00000	0.377543	0.188772	+	0.982021i	$0.439549\pi$
$450$	0	0
$451$	6.24264	0.293954
$452$	0	0
$453$	2.31371	0.108708
$454$	0	0
$455$	0	0
$456$	0	0
$457$	29.3137	1.37124	0.685619	−	0.727961i	$-0.259532\pi$
$457$	29.3137	1.37124	0.685619	+	0.727961i	$0.259532\pi$
$458$	0	0
$459$	4.82843	0.225372
$460$	0	0
$461$	13.0000	0.605470	0.302735	−	0.953075i	$-0.402100\pi$
$461$	13.0000	0.605470	0.302735	+	0.953075i	$0.402100\pi$
$462$	0	0
$463$	13.7990	0.641293	0.320647	−	0.947199i	$-0.396100\pi$
$463$	13.7990	0.641293	0.320647	+	0.947199i	$0.396100\pi$
$464$	0	0
$465$	−5.65685	−0.262330
$466$	0	0
$467$	35.9411	1.66316	0.831578	−	0.555407i	$-0.187438\pi$
$467$	35.9411	1.66316	0.831578	+	0.555407i	$0.187438\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−3.02944	−0.139589
$472$	0	0
$473$	−8.00000	−0.367840
$474$	0	0
$475$	7.75736	0.355932
$476$	0	0
$477$	−7.31371	−0.334872
$478$	0	0
$479$	14.0711	0.642923	0.321462	−	0.946923i	$-0.395826\pi$
$479$	14.0711	0.642923	0.321462	+	0.946923i	$0.395826\pi$
$480$	0	0
$481$	16.5858	0.756247
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−22.3848	−1.01644
$486$	0	0
$487$	23.3137	1.05644	0.528222	−	0.849106i	$-0.322859\pi$
$487$	23.3137	1.05644	0.528222	+	0.849106i	$0.322859\pi$
$488$	0	0
$489$	2.11270	0.0955395
$490$	0	0
$491$	0.828427	0.0373864	0.0186932	−	0.999825i	$-0.494049\pi$
$491$	0.828427	0.0373864	0.0186932	+	0.999825i	$0.494049\pi$
$492$	0	0
$493$	2.00000	0.0900755
$494$	0	0
$495$	−4.00000	−0.179787
$496$	0	0
$497$	0	0
$498$	0	0
$499$	23.7990	1.06539	0.532695	−	0.846308i	$-0.321179\pi$
$499$	23.7990	1.06539	0.532695	+	0.846308i	$0.321179\pi$
$500$	0	0
$501$	0.112698	0.00503499
$502$	0	0
$503$	−19.7279	−0.879625	−0.439812	−	0.898090i	$-0.644955\pi$
$503$	−19.7279	−0.879625	−0.439812	+	0.898090i	$0.644955\pi$
$504$	0	0
$505$	−10.5858	−0.471061
$506$	0	0
$507$	4.00000	0.177646
$508$	0	0
$509$	−35.6569	−1.58046	−0.790231	−	0.612809i	$-0.790040\pi$
$509$	−35.6569	−1.58046	−0.790231	+	0.612809i	$0.790040\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−6.24264	−0.275619
$514$	0	0
$515$	22.4853	0.990820
$516$	0	0
$517$	−4.82843	−0.212354
$518$	0	0
$519$	0.899495	0.0394834
$520$	0	0
$521$	−12.0000	−0.525730	−0.262865	−	0.964833i	$-0.584667\pi$
$521$	−12.0000	−0.525730	−0.262865	+	0.964833i	$0.584667\pi$
$522$	0	0
$523$	−13.0711	−0.571558	−0.285779	−	0.958296i	$-0.592252\pi$
$523$	−13.0711	−0.571558	−0.285779	+	0.958296i	$0.592252\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−19.3137	−0.841318
$528$	0	0
$529$	6.31371	0.274509
$530$	0	0
$531$	−12.4853	−0.541815
$532$	0	0
$533$	11.4142	0.494404
$534$	0	0
$535$	19.1716	0.828859
$536$	0	0
$537$	7.82843	0.337822
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−8.85786	−0.380829	−0.190415	−	0.981704i	$-0.560983\pi$
$541$	−8.85786	−0.380829	−0.190415	+	0.981704i	$0.560983\pi$
$542$	0	0
$543$	−5.51472	−0.236659
$544$	0	0
$545$	12.0000	0.514024
$546$	0	0
$547$	43.6985	1.86841	0.934206	−	0.356734i	$-0.116110\pi$
$547$	43.6985	1.86841	0.934206	+	0.356734i	$0.116110\pi$
$548$	0	0
$549$	−23.1127	−0.986426
$550$	0	0
$551$	−2.58579	−0.110158
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−5.31371	−0.225554
$556$	0	0
$557$	−18.8284	−0.797786	−0.398893	−	0.916997i	$-0.630606\pi$
$557$	−18.8284	−0.797786	−0.398893	+	0.916997i	$0.630606\pi$
$558$	0	0
$559$	−14.6274	−0.618674
$560$	0	0
$561$	0.828427	0.0349762
$562$	0	0
$563$	2.78680	0.117449	0.0587247	−	0.998274i	$-0.481297\pi$
$563$	2.78680	0.117449	0.0587247	+	0.998274i	$0.481297\pi$
$564$	0	0
$565$	10.5858	0.445347
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−12.9706	−0.543754	−0.271877	−	0.962332i	$-0.587644\pi$
$569$	−12.9706	−0.543754	−0.271877	+	0.962332i	$0.587644\pi$
$570$	0	0
$571$	−7.41421	−0.310275	−0.155138	−	0.987893i	$-0.549582\pi$
$571$	−7.41421	−0.310275	−0.155138	+	0.987893i	$0.549582\pi$
$572$	0	0
$573$	−2.68629	−0.112221
$574$	0	0
$575$	16.2426	0.677365
$576$	0	0
$577$	−15.9706	−0.664863	−0.332432	−	0.943127i	$-0.607869\pi$
$577$	−15.9706	−0.664863	−0.332432	+	0.943127i	$0.607869\pi$
$578$	0	0
$579$	−9.61522	−0.399595
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−2.58579	−0.107092
$584$	0	0
$585$	−7.31371	−0.302385
$586$	0	0
$587$	16.2132	0.669191	0.334595	−	0.942362i	$-0.391400\pi$
$587$	16.2132	0.669191	0.334595	+	0.942362i	$0.391400\pi$
$588$	0	0
$589$	24.9706	1.02889
$590$	0	0
$591$	4.61522	0.189845
$592$	0	0
$593$	10.5858	0.434706	0.217353	−	0.976093i	$-0.430258\pi$
$593$	10.5858	0.434706	0.217353	+	0.976093i	$0.430258\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−9.95837	−0.407569
$598$	0	0
$599$	−44.9706	−1.83745	−0.918724	−	0.394901i	$-0.870779\pi$
$599$	−44.9706	−1.83745	−0.918724	+	0.394901i	$0.870779\pi$
$600$	0	0
$601$	23.6569	0.964983	0.482492	−	0.875901i	$-0.339732\pi$
$601$	23.6569	0.964983	0.482492	+	0.875901i	$0.339732\pi$
$602$	0	0
$603$	22.8284	0.929645
$604$	0	0
$605$	−1.41421	−0.0574960
$606$	0	0
$607$	22.2843	0.904491	0.452245	−	0.891894i	$-0.350623\pi$
$607$	22.2843	0.904491	0.452245	+	0.891894i	$0.350623\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−8.82843	−0.357160
$612$	0	0
$613$	28.6274	1.15625	0.578125	−	0.815948i	$-0.303785\pi$
$613$	28.6274	1.15625	0.578125	+	0.815948i	$0.303785\pi$
$614$	0	0
$615$	−3.65685	−0.147459
$616$	0	0
$617$	33.6274	1.35379	0.676894	−	0.736080i	$-0.263325\pi$
$617$	33.6274	1.35379	0.676894	+	0.736080i	$0.263325\pi$
$618$	0	0
$619$	0.686292	0.0275844	0.0137922	−	0.999905i	$-0.495610\pi$
$619$	0.686292	0.0275844	0.0137922	+	0.999905i	$0.495610\pi$
$620$	0	0
$621$	−13.0711	−0.524524
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−1.00000	−0.0400000
$626$	0	0
$627$	−1.07107	−0.0427743
$628$	0	0
$629$	−18.1421	−0.723374
$630$	0	0
$631$	4.38478	0.174555	0.0872776	−	0.996184i	$-0.472183\pi$
$631$	4.38478	0.174555	0.0872776	+	0.996184i	$0.472183\pi$
$632$	0	0
$633$	−2.04163	−0.0811475
$634$	0	0
$635$	9.07107	0.359974
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−16.2843	−0.644196
$640$	0	0
$641$	44.2548	1.74796	0.873980	−	0.485961i	$-0.161530\pi$
$641$	44.2548	1.74796	0.873980	+	0.485961i	$0.161530\pi$
$642$	0	0
$643$	35.5858	1.40337	0.701683	−	0.712489i	$-0.252432\pi$
$643$	35.5858	1.40337	0.701683	+	0.712489i	$0.252432\pi$
$644$	0	0
$645$	4.68629	0.184523
$646$	0	0
$647$	−13.0711	−0.513877	−0.256938	−	0.966428i	$-0.582714\pi$
$647$	−13.0711	−0.513877	−0.256938	+	0.966428i	$0.582714\pi$
$648$	0	0
$649$	−4.41421	−0.173273
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−49.0711	−1.92030	−0.960150	−	0.279486i	$-0.909836\pi$
$653$	−49.0711	−1.92030	−0.960150	+	0.279486i	$0.909836\pi$
$654$	0	0
$655$	6.97056	0.272362
$656$	0	0
$657$	−9.65685	−0.376750
$658$	0	0
$659$	−31.3137	−1.21981	−0.609904	−	0.792475i	$-0.708792\pi$
$659$	−31.3137	−1.21981	−0.609904	+	0.792475i	$0.708792\pi$
$660$	0	0
$661$	4.62742	0.179986	0.0899928	−	0.995942i	$-0.471316\pi$
$661$	4.62742	0.179986	0.0899928	+	0.995942i	$0.471316\pi$
$662$	0	0
$663$	1.51472	0.0588268
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−5.41421	−0.209639
$668$	0	0
$669$	−8.10051	−0.313184
$670$	0	0
$671$	−8.17157	−0.315460
$672$	0	0
$673$	16.2426	0.626108	0.313054	−	0.949735i	$-0.398648\pi$
$673$	16.2426	0.626108	0.313054	+	0.949735i	$0.398648\pi$
$674$	0	0
$675$	−7.24264	−0.278769
$676$	0	0
$677$	−8.28427	−0.318390	−0.159195	−	0.987247i	$-0.550890\pi$
$677$	−8.28427	−0.318390	−0.159195	+	0.987247i	$0.550890\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	11.6569	0.446692
$682$	0	0
$683$	−14.8995	−0.570113	−0.285057	−	0.958511i	$-0.592012\pi$
$683$	−14.8995	−0.570113	−0.285057	+	0.958511i	$0.592012\pi$
$684$	0	0
$685$	−24.0416	−0.918583
$686$	0	0
$687$	−1.65685	−0.0632129
$688$	0	0
$689$	−4.72792	−0.180119
$690$	0	0
$691$	29.3848	1.11785	0.558925	−	0.829218i	$-0.311214\pi$
$691$	29.3848	1.11785	0.558925	+	0.829218i	$0.311214\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	4.68629	0.177761
$696$	0	0
$697$	−12.4853	−0.472914
$698$	0	0
$699$	−8.44365	−0.319368
$700$	0	0
$701$	42.4558	1.60354	0.801768	−	0.597636i	$-0.203893\pi$
$701$	42.4558	1.60354	0.801768	+	0.597636i	$0.203893\pi$
$702$	0	0
$703$	23.4558	0.884654
$704$	0	0
$705$	2.82843	0.106525
$706$	0	0
$707$	0	0
$708$	0	0
$709$	18.9289	0.710891	0.355445	−	0.934697i	$-0.384329\pi$
$709$	18.9289	0.710891	0.355445	+	0.934697i	$0.384329\pi$
$710$	0	0
$711$	22.8284	0.856133
$712$	0	0
$713$	52.2843	1.95806
$714$	0	0
$715$	−2.58579	−0.0967029
$716$	0	0
$717$	−4.85786	−0.181420
$718$	0	0
$719$	28.8284	1.07512	0.537559	−	0.843226i	$-0.319346\pi$
$719$	28.8284	1.07512	0.537559	+	0.843226i	$0.319346\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−9.41421	−0.350118
$724$	0	0
$725$	−3.00000	−0.111417
$726$	0	0
$727$	−13.7990	−0.511776	−0.255888	−	0.966706i	$-0.582368\pi$
$727$	−13.7990	−0.511776	−0.255888	+	0.966706i	$0.582368\pi$
$728$	0	0
$729$	−18.1716	−0.673021
$730$	0	0
$731$	16.0000	0.591781
$732$	0	0
$733$	−28.6569	−1.05847	−0.529233	−	0.848477i	$-0.677520\pi$
$733$	−28.6569	−1.05847	−0.529233	+	0.848477i	$0.677520\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	8.07107	0.297302
$738$	0	0
$739$	−36.4264	−1.33997	−0.669984	−	0.742376i	$-0.733699\pi$
$739$	−36.4264	−1.33997	−0.669984	+	0.742376i	$0.733699\pi$
$740$	0	0
$741$	−1.95837	−0.0719425
$742$	0	0
$743$	−34.0000	−1.24734	−0.623670	−	0.781688i	$-0.714359\pi$
$743$	−34.0000	−1.24734	−0.623670	+	0.781688i	$0.714359\pi$
$744$	0	0
$745$	22.6274	0.829004
$746$	0	0
$747$	29.6569	1.08509
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−11.3137	−0.412843	−0.206422	−	0.978463i	$-0.566182\pi$
$751$	−11.3137	−0.412843	−0.206422	+	0.978463i	$0.566182\pi$
$752$	0	0
$753$	−0.201010	−0.00732522
$754$	0	0
$755$	7.89949	0.287492
$756$	0	0
$757$	17.3137	0.629277	0.314639	−	0.949212i	$-0.398117\pi$
$757$	17.3137	0.629277	0.314639	+	0.949212i	$0.398117\pi$
$758$	0	0
$759$	−2.24264	−0.0814027
$760$	0	0
$761$	27.9411	1.01287	0.506433	−	0.862280i	$-0.330964\pi$
$761$	27.9411	1.01287	0.506433	+	0.862280i	$0.330964\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	8.00000	0.289241
$766$	0	0
$767$	−8.07107	−0.291429
$768$	0	0
$769$	−30.9706	−1.11683	−0.558414	−	0.829563i	$-0.688590\pi$
$769$	−30.9706	−1.11683	−0.558414	+	0.829563i	$0.688590\pi$
$770$	0	0
$771$	−8.07107	−0.290672
$772$	0	0
$773$	−34.6274	−1.24546	−0.622731	−	0.782436i	$-0.713977\pi$
$773$	−34.6274	−1.24546	−0.622731	+	0.782436i	$0.713977\pi$
$774$	0	0
$775$	28.9706	1.04065
$776$	0	0
$777$	0	0
$778$	0	0
$779$	16.1421	0.578352
$780$	0	0
$781$	−5.75736	−0.206015
$782$	0	0
$783$	2.41421	0.0862770
$784$	0	0
$785$	−10.3431	−0.369163
$786$	0	0
$787$	5.61522	0.200161	0.100081	−	0.994979i	$-0.468090\pi$
$787$	5.61522	0.200161	0.100081	+	0.994979i	$0.468090\pi$
$788$	0	0
$789$	−9.54416	−0.339781
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−14.9411	−0.530575
$794$	0	0
$795$	1.51472	0.0537215
$796$	0	0
$797$	−9.31371	−0.329908	−0.164954	−	0.986301i	$-0.552748\pi$
$797$	−9.31371	−0.329908	−0.164954	+	0.986301i	$0.552748\pi$
$798$	0	0
$799$	9.65685	0.341635
$800$	0	0
$801$	−25.9411	−0.916585
$802$	0	0
$803$	−3.41421	−0.120485
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−5.65685	−0.199131
$808$	0	0
$809$	−8.00000	−0.281265	−0.140633	−	0.990062i	$-0.544914\pi$
$809$	−8.00000	−0.281265	−0.140633	+	0.990062i	$0.544914\pi$
$810$	0	0
$811$	49.6569	1.74369	0.871844	−	0.489784i	$-0.162924\pi$
$811$	49.6569	1.74369	0.871844	+	0.489784i	$0.162924\pi$
$812$	0	0
$813$	0.0294373	0.00103241
$814$	0	0
$815$	7.21320	0.252667
$816$	0	0
$817$	−20.6863	−0.723722
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−22.1716	−0.773793	−0.386897	−	0.922123i	$-0.626453\pi$
$821$	−22.1716	−0.773793	−0.386897	+	0.922123i	$0.626453\pi$
$822$	0	0
$823$	27.1716	0.947141	0.473571	−	0.880756i	$-0.342965\pi$
$823$	27.1716	0.947141	0.473571	+	0.880756i	$0.342965\pi$
$824$	0	0
$825$	−1.24264	−0.0432632
$826$	0	0
$827$	−32.5269	−1.13107	−0.565536	−	0.824724i	$-0.691331\pi$
$827$	−32.5269	−1.13107	−0.565536	+	0.824724i	$0.691331\pi$
$828$	0	0
$829$	−37.4142	−1.29945	−0.649725	−	0.760170i	$-0.725116\pi$
$829$	−37.4142	−1.29945	−0.649725	+	0.760170i	$0.725116\pi$
$830$	0	0
$831$	10.2721	0.356334
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0.384776	0.0133157
$836$	0	0
$837$	−23.3137	−0.805840
$838$	0	0
$839$	26.4853	0.914373	0.457187	−	0.889371i	$-0.348857\pi$
$839$	26.4853	0.914373	0.457187	+	0.889371i	$0.348857\pi$
$840$	0	0
$841$	−28.0000	−0.965517
$842$	0	0
$843$	4.92893	0.169761
$844$	0	0
$845$	13.6569	0.469810
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−11.4142	−0.391735
$850$	0	0
$851$	49.1127	1.68356
$852$	0	0
$853$	50.0000	1.71197	0.855984	−	0.517003i	$-0.172952\pi$
$853$	50.0000	1.71197	0.855984	+	0.517003i	$0.172952\pi$
$854$	0	0
$855$	−10.3431	−0.353728
$856$	0	0
$857$	48.2843	1.64936	0.824680	−	0.565600i	$-0.191355\pi$
$857$	48.2843	1.64936	0.824680	+	0.565600i	$0.191355\pi$
$858$	0	0
$859$	−12.4142	−0.423568	−0.211784	−	0.977317i	$-0.567927\pi$
$859$	−12.4142	−0.423568	−0.211784	+	0.977317i	$0.567927\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	26.3848	0.898148	0.449074	−	0.893495i	$-0.351754\pi$
$863$	26.3848	0.898148	0.449074	+	0.893495i	$0.351754\pi$
$864$	0	0
$865$	3.07107	0.104419
$866$	0	0
$867$	5.38478	0.182877
$868$	0	0
$869$	8.07107	0.273792
$870$	0	0
$871$	14.7574	0.500034
$872$	0	0
$873$	−44.7696	−1.51522
$874$	0	0
$875$	0	0
$876$	0	0
$877$	13.2843	0.448578	0.224289	−	0.974523i	$-0.427994\pi$
$877$	13.2843	0.448578	0.224289	+	0.974523i	$0.427994\pi$
$878$	0	0
$879$	−5.45584	−0.184021
$880$	0	0
$881$	−19.8284	−0.668037	−0.334018	−	0.942567i	$-0.608405\pi$
$881$	−19.8284	−0.668037	−0.334018	+	0.942567i	$0.608405\pi$
$882$	0	0
$883$	−39.6690	−1.33497	−0.667485	−	0.744623i	$-0.732629\pi$
$883$	−39.6690	−1.33497	−0.667485	+	0.744623i	$0.732629\pi$
$884$	0	0
$885$	2.58579	0.0869203
$886$	0	0
$887$	−31.7279	−1.06532	−0.532660	−	0.846330i	$-0.678807\pi$
$887$	−31.7279	−1.06532	−0.532660	+	0.846330i	$0.678807\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−7.48528	−0.250766
$892$	0	0
$893$	−12.4853	−0.417804
$894$	0	0
$895$	26.7279	0.893416
$896$	0	0
$897$	−4.10051	−0.136912
$898$	0	0
$899$	−9.65685	−0.322074
$900$	0	0
$901$	5.17157	0.172290
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−18.8284	−0.625878
$906$	0	0
$907$	12.6274	0.419286	0.209643	−	0.977778i	$-0.432770\pi$
$907$	12.6274	0.419286	0.209643	+	0.977778i	$0.432770\pi$
$908$	0	0
$909$	−21.1716	−0.702217
$910$	0	0
$911$	−38.4853	−1.27507	−0.637537	−	0.770420i	$-0.720047\pi$
$911$	−38.4853	−1.27507	−0.637537	+	0.770420i	$0.720047\pi$
$912$	0	0
$913$	10.4853	0.347012
$914$	0	0
$915$	4.78680	0.158247
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−5.45584	−0.179972	−0.0899858	−	0.995943i	$-0.528682\pi$
$919$	−5.45584	−0.179972	−0.0899858	+	0.995943i	$0.528682\pi$
$920$	0	0
$921$	−3.55635	−0.117186
$922$	0	0
$923$	−10.5269	−0.346498
$924$	0	0
$925$	27.2132	0.894765
$926$	0	0
$927$	44.9706	1.47703
$928$	0	0
$929$	41.4264	1.35916	0.679578	−	0.733603i	$-0.262163\pi$
$929$	41.4264	1.35916	0.679578	+	0.733603i	$0.262163\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	11.2721	0.369031
$934$	0	0
$935$	2.82843	0.0924995
$936$	0	0
$937$	21.4142	0.699572	0.349786	−	0.936830i	$-0.386254\pi$
$937$	21.4142	0.699572	0.349786	+	0.936830i	$0.386254\pi$
$938$	0	0
$939$	−3.87006	−0.126295
$940$	0	0
$941$	50.9411	1.66063	0.830317	−	0.557292i	$-0.188160\pi$
$941$	50.9411	1.66063	0.830317	+	0.557292i	$0.188160\pi$
$942$	0	0
$943$	33.7990	1.10065
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−0.201010	−0.00653195	−0.00326598	−	0.999995i	$-0.501040\pi$
$947$	−0.201010	−0.00653195	−0.00326598	+	0.999995i	$0.501040\pi$
$948$	0	0
$949$	−6.24264	−0.202645
$950$	0	0
$951$	−8.68629	−0.281672
$952$	0	0
$953$	17.6152	0.570613	0.285307	−	0.958436i	$-0.407905\pi$
$953$	17.6152	0.570613	0.285307	+	0.958436i	$0.407905\pi$
$954$	0	0
$955$	−9.17157	−0.296785
$956$	0	0
$957$	0.414214	0.0133896
$958$	0	0
$959$	0	0
$960$	0	0
$961$	62.2548	2.00822
$962$	0	0
$963$	38.3431	1.23559
$964$	0	0
$965$	−32.8284	−1.05678
$966$	0	0
$967$	18.3431	0.589876	0.294938	−	0.955516i	$-0.404701\pi$
$967$	18.3431	0.589876	0.294938	+	0.955516i	$0.404701\pi$
$968$	0	0
$969$	2.14214	0.0688153
$970$	0	0
$971$	30.2132	0.969588	0.484794	−	0.874628i	$-0.338895\pi$
$971$	30.2132	0.969588	0.484794	+	0.874628i	$0.338895\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−2.27208	−0.0727647
$976$	0	0
$977$	3.02944	0.0969203	0.0484601	−	0.998825i	$-0.484569\pi$
$977$	3.02944	0.0969203	0.0484601	+	0.998825i	$0.484569\pi$
$978$	0	0
$979$	−9.17157	−0.293125
$980$	0	0
$981$	24.0000	0.766261
$982$	0	0
$983$	−29.5147	−0.941373	−0.470687	−	0.882300i	$-0.655994\pi$
$983$	−29.5147	−0.941373	−0.470687	+	0.882300i	$0.655994\pi$
$984$	0	0
$985$	15.7574	0.502071
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−43.3137	−1.37730
$990$	0	0
$991$	−0.100505	−0.00319265	−0.00159632	−	0.999999i	$-0.500508\pi$
$991$	−0.100505	−0.00319265	−0.00159632	+	0.999999i	$0.500508\pi$
$992$	0	0
$993$	−2.23045	−0.0707811
$994$	0	0
$995$	−34.0000	−1.07787
$996$	0	0
$997$	−41.4558	−1.31292	−0.656460	−	0.754361i	$-0.727947\pi$
$997$	−41.4558	−1.31292	−0.656460	+	0.754361i	$0.727947\pi$
$998$	0	0
$999$	−21.8995	−0.692869

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8624.2.a.cd.1.1		2
4.3	odd	2		4312.2.a.m.1.2		2
7.3	odd	6		1232.2.q.h.177.1		4
7.5	odd	6		1232.2.q.h.529.1		4
7.6	odd	2		8624.2.a.bg.1.2		2
28.3	even	6		616.2.q.b.177.2	✓	4
28.19	even	6		616.2.q.b.529.2	yes	4
28.27	even	2		4312.2.a.u.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
616.2.q.b.177.2	✓	4	28.3	even	6
616.2.q.b.529.2	yes	4	28.19	even	6
1232.2.q.h.177.1		4	7.3	odd	6
1232.2.q.h.529.1		4	7.5	odd	6
4312.2.a.m.1.2		2	4.3	odd	2
4312.2.a.u.1.1		2	28.27	even	2
8624.2.a.bg.1.2		2	7.6	odd	2
8624.2.a.cd.1.1		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 8624 = 2^{4} \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8624.a (trivial)

\(f(q)\)	\(=\)	\(q-0.414214 q^{3} -1.41421 q^{5} -2.82843 q^{9} -1.00000 q^{11} -1.82843 q^{13} +0.585786 q^{15} +2.00000 q^{17} -2.58579 q^{19} -5.41421 q^{23} -3.00000 q^{25} +2.41421 q^{27} +1.00000 q^{29} -9.65685 q^{31} +0.414214 q^{33} -9.07107 q^{37} +0.757359 q^{39} -6.24264 q^{41} +8.00000 q^{43} +4.00000 q^{45} +4.82843 q^{47} -0.828427 q^{51} +2.58579 q^{53} +1.41421 q^{55} +1.07107 q^{57} +4.41421 q^{59} +8.17157 q^{61} +2.58579 q^{65} -8.07107 q^{67} +2.24264 q^{69} +5.75736 q^{71} +3.41421 q^{73} +1.24264 q^{75} -8.07107 q^{79} +7.48528 q^{81} -10.4853 q^{83} -2.82843 q^{85} -0.414214 q^{87} +9.17157 q^{89} +4.00000 q^{93} +3.65685 q^{95} +15.8284 q^{97} +2.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 2 q^{11} + 2 q^{13} + 4 q^{15} + 4 q^{17} - 8 q^{19} - 8 q^{23} - 6 q^{25} + 2 q^{27} + 2 q^{29} - 8 q^{31} - 2 q^{33} - 4 q^{37} + 10 q^{39} - 4 q^{41} + 16 q^{43} + 8 q^{45} + 4 q^{47}+ \cdots + 26 q^{97}+O(q^{100}) \) 2 * q + 2 * q^3 - 2 * q^11 + 2 * q^13 + 4 * q^15 + 4 * q^17 - 8 * q^19 - 8 * q^23 - 6 * q^25 + 2 * q^27 + 2 * q^29 - 8 * q^31 - 2 * q^33 - 4 * q^37 + 10 * q^39 - 4 * q^41 + 16 * q^43 + 8 * q^45 + 4 * q^47 + 4 * q^51 + 8 * q^53 - 12 * q^57 + 6 * q^59 + 22 * q^61 + 8 * q^65 - 2 * q^67 - 4 * q^69 + 20 * q^71 + 4 * q^73 - 6 * q^75 - 2 * q^79 - 2 * q^81 - 4 * q^83 + 2 * q^87 + 24 * q^89 + 8 * q^93 - 4 * q^95 + 26 * q^97

Introduction

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